Let $n \in \mathbb{N}$ and $A\in M_n(\mathbb{N})$ with $Tr(A)=0$ and $A^3+A-2I_n=O_n$.Prove that $n$ is a multiple of $3$ and $\det(A^2)=\det(A^2+I_n)$.
I tried to find $A$'s eigenvalues, but the equation $x^3+x-2=0$ has only one integer root, $1$,and this contradicts $Tr(A) =0$.
EDIT:My approach doesn' t work as pointed out in the comments, how should this be solved?

$\begingroup$Eigenvalues need not only be integers.$\endgroup$
– Theo BenditJan 11 at 15:19

$\begingroup$Why not? The matrix is over $N$. I know that if it were over $C$ it would have eigenvalues.$\endgroup$
– JustAnAmateurJan 11 at 15:20

$\begingroup$The eigenvalues of a matrix with entries in, for instance $\mathbb N$, are solutions to a polynomial equation whose coefficients are in $\mathbb N$. These solutions need not (and in fact are often not) in $\mathbb N$.$\endgroup$
– DaveJan 11 at 15:21

2 Answers
2

Since $A^3 + A - 2 I = 0$, all eigenvalues of $A$ are roots of the polynomial $x^2 + x - 2$, thus $1$ or $-1/2 \pm \sqrt{7} i/2$. Since it's a real matrix, the two non-real eigenvalues have equal algebraic multiplicities. Since the trace (which is the sum of the eigenvalues) must be $0$,
all three eigenvalues have equal multiplicities. Thus if this multiplicity is $k$,
there are $3k$ eigenvalues counted by algebraic multiplicity, i.e. $n=3k$.

$\begingroup$Thank you for your solution ! I only have one question : you said this is a real matrix, but the problem states it is a positive integer matrix. Are we still allowed to use the fact that the trace is the sum of the eigenvalues? I know that this works only over algebraically closed fields.$\endgroup$
– JustAnAmateurJan 12 at 12:01

$\begingroup$The matrix might have entries in $\mathbb N$, but the eigenvalues are complex numbers. Yes, this is allowed.$\endgroup$
– Robert IsraelJan 13 at 6:46

Take the matrix in $GL_2(\Bbb Z)$$$
\begin{pmatrix}
1 & 1 \\
1 & 0
\end{pmatrix},
$$
it has not integers as eigenvalues, but the golden ratio $\frac{1\pm \sqrt{5}}{2} $. If you want to construct matrices with integer eigenvalues, then see for example here: